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Ball-screw axis moving 50 kg by 0.5 m in 0.6 s: reflected inertia, 1/3-1/3-1/3 profile, peak and RMS torque and three candidate servo motors

Tool	Motor and Drive Sizing
Source	U.S. Army Materiel Command (1963). Engineering Design Handbook: Servomechanisms, Section 4, Power Elements and System Design. AMCP 706-139, Headquarters, U.S. Army Materiel Command, Washington, DC, par. 15-1.13 (reflected inertia), eqs. 14-2 and 14-19 (DC machine), par. 16-3 and its appendix (choice of motor and gear ratio). The axis itself is illustrative, not a printed example.
Level	Professional

Problem

A 50 kg table on rolling guides (μ = 0.01) is driven directly by a ball screw of 20 mm lead and 0.9 efficiency; the steel screw is 20 mm in diameter and 800 mm long, the coupling adds 0.00002 kg·m² and the motor shaft has 0.05 N·m of drag. The table moves 500 mm in 0.6 s by the 1/3-1/3-1/3 trapezoid and then dwells 0.4 s. Find the reflected inertia, the peak speed, the torque of each segment, the peak and RMS torque and the peak power, and check three candidate motors (200 W, 400 W, 750 W) for torque, speed, inertia ratio, current, voltage on a 320 V bus and winding temperature rise.

Inputs to Enter

Mechanism: lead or ball screw; load 50 kg, μ 0.01, no external force, level; ratio 1, screw lead 20 mm, screw efficiency 0.9
Motor-side inertia 0.00002 kg·m²; screw as a steel cylinder 20 mm × 800 mm (7,850 kg/m³); drag torque 0.05 N·m
Profile 1/3-1/3-1/3: 500 mm in 0.6 s, dwell 0.4 s, constant acceleration, one move per cycle, motor holds at rest
Inertia ratio limit 10; bus 320 V, bus limit 390 V
Motors (J, rated / peak torque, max speed, K_t, K_e, R, R_th): 200 W: 0.000015 kg·m², 0.64 / 1.91 N·m, 6,000 rpm, 0.35 N·m/A, 36.7 V/krpm, 4 Ω, 2 °C/W; 400 W: 0.000028, 1.27 / 3.82, 6,000, 0.45, 47.1, 2, 1.5; 750 W: 0.00011, 2.39 / 7.16, 5,000, 0.5, 52.4, 1, 1

Open the Motor and Drive Sizing tool (loaded with this example)

Answers

Screw J = π ρ L d⁴/32 = 9.865×10⁻⁵ kg·m²; load m (lead/2π)² = 5.066×10⁻⁴ kg·m²; reflected load inertia 6.253×10⁻⁴ kg·m²
Peak speed 1.5 X/t = 1.25 m/s = 3,750 rpm at the motor; acceleration 4.5 X/t² = 6.25 m/s² (0.637 g)
750 W motor: accelerate 1.622 N·m, cruise 0.0673 N·m, decelerate −1.280 N·m, dwell 0; T_pk = 1.622 N·m, T_rms = 0.9244 N·m, P_pk = 636.8 W
750 W: inertia ratio 5.68, T_pk / peak 0.226, T_rms / rated 0.387, speed 0.750: passes; I_pk 3.243 A, I_rms 1.849 A, 199.7 V needed, winding rise 3.4 °C
400 W: T_rms 0.8234 N·m (0.648 of rated): passes, but inertia ratio 22.3 exceeds 10; 200 W: T_rms 0.8074 N·m against 0.64 N·m rated: fails on RMS torque
Ratio of least ramp torque for the 750 W motor N_opt = 2.173; AMCP 706-139 window R = 0.169 to 1.333 (speed limit); 49.0 J returned to the bus per deceleration, 1,972 µF to hold it between 320 and 390 V

Notes

The tool's default inputs are this axis; press Submit. The axis and the motor data are illustrative round numbers, not catalogue values. Checked by test_motor_drive_sizing.php (784 checks) against an independent numpy / scipy integration of the cycle and hand build-ups of each torque; the handbook's printed servo example (p. 16-21: 5.56 rad/s and 11.7 rad/s²) is reproduced in the same test. The default page also runs an induction-motor start (15 kW, 1,750 rpm, fan load): full-load torque 81.85 N·m, starting time 3.958 s.